Algebra and Graphing Calculators

Calculators are widely used in classrooms today to support student learning of mathematics.  Ten years ago, Dion et al. (2001) conducted a wide-spread survey of school calculator use.  Of the 4,568 schools that responded, 95% answered that they either require (46%) or at least allow (49%) the use of graphing or scientific calculators.

Both the Common Core State Standards in Mathematics (CCSS-M) and the National Council of Teachers of Mathematics (NCTM) Standards make strong recommendations for the use of technology in the classroom. Technology should be a tool “in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning” (NCTM, 2000, p. 24). The Common Core State Standards (2010) Standards for Mathematical Practice states, as an example of student tool use, that “mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator.  They detect possible errors by strategically using estimation and other mathematical knowledge” (p. 7).

The widespread use of calculators in the classroom is not necessarily equivalent to widespread effective use.  Milou (1999) surveyed 146 teachers and concluded that “successful integration of the graphing calculator into the mathematics classroom cannot take place without the aid of enthusiastic teachers.”

Both the NCTM and CCSS-M standards previously quoted encourage the use of technology as tools, when appropriate.  Hiebert et al. (1997)7) stated: ‘Students must construct meaning for all tools…. As you use a tool, you get to know the tool better and you use the tool more effectively to help you know about other things.’ (p. 54). This suggests that teachers and students both must learn to use technology such as graphing calculators by co-constructing that knowledge.

Calculators as Tools not “Dispensers of Answers”

In 1981 a study was conducted to study the process of student thinking in estimation tasks.  As one part of the study, the researchers asked 33 participants of various ages from child to adult, who had been identified as the top 10% of estimators in previous interviews, to estimate a calculation and then to use a calculator to check the answer.  They used a specially programmed HP-65 calculator that would make increasingly large errors in calculation.  In the study, 36% of females and 77% of males said something about the calculator being in error before they reached the last problem. One conclusion drawn by the researchers was that: “[t]he unwillingness of these good estimators to reject unreasonable answers suggests that a challenging task lies ahead in preparing students to be alert to unreasonable answers” (Reys, 1980, p. 231).

This study was redone by Glasgow and Reys (1998) almost 20 years later with 25 undergraduate participants each of whom had been identified as a top student in his or mathematics course.  In this second experiment, only 28% of the participants questioned the calculator before the last question.  Several of the participants in this second study were pre-service teachers.  One statement by the authors in the conclusion mentioned the importance of convincing these pre-service teachers “to question or reflect on their own mathematical thinking”.  The authors also argued that students needed to see calculators “as tools, not ‘dispensers of answers’” (p. 388).

In order to be able to use the calculator effectively in the classroom to support student learning of effective use of this tool rather than dependence on it, teachers must first construct their own knowledge of the calculator as a tool rather than as a sort of magic or “black box” that is unconnected to reality and is unpredictable.

Knowledge for Teachers

Chamblee, Slough, and Wunsch (2010) conducted a study to measure the concerns of in-service teachers before and after a year-long professional development program.  This program intended to support mathematics and science teachers in their use of graphing calculators, probes and sensors in the classroom.  90 teachers received 105 days of professional development throughout the school year and following summer.  The teachers also received a classroom set of graphing calculators, probes and sensors. The researchers tested 22 teachers before and after 105 days of professional development.  One of their findings was teachers needed professional development activities that allow them to learn how to use the graphing calculator in their own day-to-day classroom, instead of watching the professional development facilitator “show-and-tell” possible uses and activities.

This seems to suggest that seeing the calculator and its functionality is not enough to incorporate it into classroom life. The teacher needs opportunities to meaningfully construct her own knowledge – she needs to be involved in her own learning process in order to construct the knowledge necessary to strategically use the calculator in the classroom.

Support for Student Learning

In Doerr and Zangor (2000) this co-construction of the graphing calculator as a tool was studied in two pre-calculus classes all taught by the same teacher. The students used graphing calculators in class, along with probes and sensors for data collection.  The researchers suggested that both teachers and students can use a graphing calculator in different ways: as a computational tool, transformational tool, data collection and analysis tool, visualizing tool, or checking tool (p. 151).  As a computational tool, it is merely used to evaluate numerical expressions.  The danger is that students will see it only as a computational tool (Herman, 2007; Ruthven, 1995).  As a transformation tool, the calculator transforms “tedious computational tasks… into interpretative tasks” (p. 153).  In support of this role, the teacher must be often re-focusing the students’ attention away from the computation and to interpretation.  As a checking tool, the students might use the calculator to verify results acquired through different methods.  One danger of allowing students to use the calculator too often as a checking tool might be that it would lead to the students abdicating responsibility to their calculators, as discussed above in Reys (1980) and Glasgow and Reys (1998).

Pugalee (2001) also suggested that graphing calculators are not a magic device that result in student learning.  He argued that technology must be used with intentional discourse in order to be effective.  He performed a study using constructivism and graphing calculators in an algebra course for “at-risk” students.  The study covered only one section of 16 students, but the results were favorable. The use of graphing calculators, along with questioning, supported the students’ construction of mathematical knowledge. These results support Doerr and Zangor (2000) above since they also argued that the teacher needs to focus the students’ attention on interpreting calculator results instead of only focusing on the calculation in order to increase their mathematical knowledge.

References

CCSS-I. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Chamblee, G. E., Slough, S. W., & Wunsch, G. (2010). Measuring High School Mathematics Teachers’ Concerns About Graphing Calculators and Change: A Year Long Study. Journal of Computers in Mathematics and Science Teaching, 27(2), 183.

Dion, G., Harvey, A., Jackson, C., Klag, P., Liu, J., & Wright, C. (2001). A survey of calculator usage in high schools. School Science and Mathematics, 101(8), 427-438.

Doerr, H. M., & Zangor, R. (2000). Creating meaning for and with the graphing calculator. Educational Studies in Mathematics, 41(2), 143-163.

Glasgow, B., & Reys, B. J. (1998). The authority of the calculator in the minds of college students. School Science and Mathematics, 98(7), 383-388.

Harvey, J. G., Waits, B. K., & Demana, F. D. (1995). The influence of technology on the teaching and learning of algebra. The Journal of Mathematical Behavior, 14(1), 75-109.

Herman, M. (2007). What Students Choose to Do and Have to Say About Use of Multiple Representations in College Algebra. The Journal of Computers in Mathematics and Science Teaching, 26(1), 27-54.

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Wearne, D., Murray, H., . . . Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.

Hollar, J. C., & Norwood, K. (1999). The effects of a graphing-approach intermediate algebra curriculum on students’ understanding of function. Journal for Research in Mathematics Education, 30(2), 220-220.

Mayes, R. L. (1995). The application of a computer algebra system as a tool in college algebra. School Science and Mathematics, 95(2), 61-68.

Milou, E. (1999). The Graphing Calculator: A Survey of Classroom Usage. School Science and Mathematics, 99(3).

NCTM. (2000). Principles and Standards for School Mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc.

O’Callaghan, B. R. (1998). Computer-intensive algebra and students’ conceptual knowledge of functions. Journal for Research in Mathematics Education, 21-40.

Pugalee, D. K. (2001). Algebra for all: The role of technology and constructivism in an algebra course for at-risk students. Preventing School Failure, 45(4), 171-176.

Pullano, F. (2005). Using Probeware to Improve Students’ Graph Interpretation Abilities. School Science and Mathematics, 105(7), 373-376.

Reys, R. E. (1980). Identification and Characterization of Computational Estimation Processes Used by Inschool Pupils and Out-of-School Adults. Final Report.

Ruthven, K. (1995). Pupil’s View of Number Work and Calculators. Educational Research, 37, 229-237.

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